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Total blow-ups of modules and universal flatifications
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-8893-5211
2017 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 45, no 9, 3706-3715 p.Article in journal (Refereed) Published
Abstract [en]

We study the projective spectrum of the Rees algebra of a module, and characterize it by a universal property. As applications, we give descriptions of universal flatifications of modules and of birational projective morphisms.

Place, publisher, year, edition, pages
Taylor & Francis, 2017. Vol. 45, no 9, 3706-3715 p.
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195714DOI: 10.1080/00927872.2016.1244270ISI: 000399458900004Scopus ID: 2-s2.0-85009957886OAI: oai:DiVA.org:kth-195714DiVA: diva2:1045290
Funder
Swedish Research Council, 2011-5599
Note

QCR 20161110

QC 20170602

Available from: 2016-11-08 Created: 2016-11-08 Last updated: 2017-06-02Bibliographically approved
In thesis
1. Hilbert schemes and Rees algebras
Open this publication in new window or tab >>Hilbert schemes and Rees algebras
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field.

The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property.

The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. vii, 49 p.
Series
TRITA-MAT-A, 2016:11
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-195717 (URN)978-91-7729-171-8 (ISBN)
Public defence
2016-12-08, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20161110

Available from: 2016-11-10 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
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